"Davul dengi dengine diye çalar"

This is a tricky one.

Davul is a big drum....correct
Comak is the stick you hit the drum with, to play.

Although no reference is made to the comak here, what is meant is that drum sounds best when it is in perfect balance with its comak. (you cant play a huge drum, with a 10cm stick, or vica versa)

Inference is that the same principle applies in interpersonal relations: Balance, between partners, is required for good music.

Aferin AlphaF, güzel bir açıklama!
Well done , nice explanation

Thanks - I enjoyed learning this!

So it doesn't match "Birds of a feather flock together" which is really often a negative type of statement that people tend to hang round with people like them .
e.g. she is a liar, and so she makes friends with other people who are also liars. or it is no wonder she is a liar when you look at who her friends are.

I couldn't think of a saying in English that matched "marriages work best when you are equally balanced", just a quote from the Bible that says "do not be unequally yoked" (image of two oxen in a plough: if one os big and the other small they don't work well together).

Can anybody think of one?

Well, Marionin!
what do you think about a situation when someone choosing a liar to marry with? If she or he stepping forward to realize that marriage then he or she enjoying that and thought it is according to her/his own standards. But this is the dark side of the issue.
The other side is we have to be careful while choosing and look for the best.